Slow–fast systems and sliding on codimension 2 switching manifolds

ABSTRACT

In this work, we consider piecewise smooth vector fields X defined in ${\mathbb{R}}^n\setminus \mathrm{\Sigma }$, where Σ is a self-intersecting switching manifold. A double regularization of X is a 2-parameter family of smooth vector fields $X_{\epsilon .\eta }$, $\epsilon ,\eta >0,$ satisfying that $X_{\epsilon ,\eta }$ converges uniformly to X in each compact subset of ${\mathbb{R}}^n\mathrm{\setminus \Sigma }$ when $\epsilon ,\eta \to 0$. We define the sliding region on the non-regular part of Σ as a limit of invariant manifolds of $X_{\epsilon .\eta }$. Since the double regularization provides a slow–fast system, the GSP-theory (geometric singular perturbation theory) is our main tool.

KEYWORDS:

• Singular perturbation
• non-smooth systems
• invariant manifolds
• Hopf bifurcation
• Bogdanov–Takens bifurcation

2010 MATHEMATICS SUBJECT CLASSIFICATION 2010:

• 34D15
• 34C45
• 34C07
• 34C23
• 34C25

Acknowledgments

The authors are grateful for the suggestions and comments of Daniel Cantergiani Panazzolo and for the hospitality of LMIA-UNIVERSITÉ DE HAUTE-ALSACE.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 As usual, we denote $Xf=\mathrm{\nabla }f.X$.

2 By definition, this is a $C^{\mathrm{\infty }}$ function such that $\varphi \left(t\right)=-1$ for $t\le -1$, $\varphi \left(t\right)=1$ for $t\ge 1$ and ${\varphi }^{\prime }\left(t\right)>0$ for $-1<t<1$.

Additional information

Funding

The authors are partially supported by CAPES/Brazil (program PROCAD) [grant number 88881.068462/2014-01] and FAPESP [grant number 2013/24541-0].